The reason you need that letter is because Social Services requires a lease as evidence of the expense. The receipts do not necessarily show that you will continue to have the same cost for rent in the future, and Social Services does not want to provide benefits to those who cannot show that the amount will be payable in the future.
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The food stamp office is asking for the landlord's letter specifically because you don't have a lease. If you have submitted all of that other proof of residence, you can try asking the food stamp office to drop the landlord's letter request.
You should not be afraid to ask the landlord for the letter. You are not asking for a lease, just an acknowledgement that you and your family live there. Landlords in New York City provide these letters for their tenants to present to HRA -- whether or not those tenants have leases -- all of the time.